- Rosenlicht. Chap. 7, exercises 22-24. In #22, I suggest you deal first with the case in which c0 is not 0. Make sure you give a proof, not a hand-wave. Failure to show that you realize where the hypotheses are being used will be penalized. In #24, include the argument for the "hence that ...". Due date: Fri. 3/20/09.
- Improper Integrals handout, exercises 17-19. Due date: Fri. 3/27/09. Hint for #17: relate the integral to an alternating series. Drawing a picture may help you figure out how to do this.
- Rosenlicht. Chap. 7, exercises 25,26,28-31,35,36. Due date: Mon. 3/30/09.
- Rosenlicht. Chap. 7, exercises 37,39,40. In #39, you'll use parts (c) and (d) to deduce part (e); don't assume part (e) in order to do part (d). Hint for part (d): start with the inequality I2n+2 < I2n+1 < I2n (from the first assertion in part (d)), divide by I2n, and use part (b). Due date: Wed. 4/1/09.
Note 1: In #39 and #40, if you're unable to do one of the parts, you are allowed to assume it for the purpose of doing subsequent parts (and similarly you may assume 39(e) in order to do 40(d)). You are not permitted to assume later parts to do earlier parts; this would defeat the purpose of the problems. In particular, do not assume 39(e) in order to do 39(d).
Note 2: Anyone who tells me that the limit in 39(d) is 1 because the numerator and denominator have the same limit, or because "0/0=1", risks meeting an untimely end and should avoid dark alleys.
Note 3: The formula in 40(d) doesn't drop as quickly out of 40(c) and 39(e) as Rosenlicht's efficient wording of the strategy might lead you to hope.
Note 4: Appreciate what 40(d) is telling you: it gives you a closed-form, non-recursive estimate of n! that gets arbitrarily good as n goes to infinity. Specifically, Stirling's formula says that for large n, n! is approximately {(n/e)n times the square root of 2πn}. This answers the question "Just how big is n! in terms of power and exponential functions?"